On the singular local limit for conservation laws with nonlocal fluxes

TL;DR

This paper investigates the rigorous justification of the local limit of nonlocal conservation laws with solution-dependent velocities, showing convergence under viscous perturbations but not in the purely nonlocal case.

Contribution

It provides counterexamples to convergence in the nonlocal case and proves convergence to viscous conservation laws when viscous perturbations are included.

Findings

Counterexamples show non-convergence in the nonlocal limit.

Viscous perturbations ensure convergence to viscous conservation laws.

Numerical evidence does not always imply rigorous convergence.

Abstract

We give an answer to a question posed in [P. Amorim, R. Colombo, and A. Teixeira, ESAIM Math. Model. Numerics. Anal. 2015], which can be loosely speaking formulated as follows. Consider a family of continuity equations where the velocity depends on the solution via the convolution by a regular kernel. In the singular limit where the convolution kernel is replaced by a Dirac delta, one formally recovers a conservation law: can we rigorously justify this formal limit? We exhibit counterexamples showing that, despite numerical evidence suggesting a positive answer, one in general does not have convergence of the solutions. We also show that the answer is positive if we consider viscous perturbations of the nonlocal equations. In this case, in the singular local limit the solutions converge to the solution of the viscous conservation law.